Going on the assumption that your MacBook Air won't "start" means it won't turn on I will assume you can't boot into Target Disk Mode. So, to rephrase your question, Can you remove the drive from the dead MacBook and connect it to the functioning Mac to transfer the data?
The answer is yes. You will need to basically do two things:
(You didn't specify which MacBook Air you have and it is very important as the drives are very different).
Removing the SSD
The procedure for removing the SSD for all MBAs is essentially the same - you remove the back cover, remove the retaining screws of the SSD and disconnect the SSD module.
Ifixt.com has excellent instructions and this procedure is rated as "moderate" (no idea why, it's super easy) and only 5-10 mins to complete.
Install in USB Enclosure
Next, you install it into the USB enclosure - just plug it in - it only goes in one way. Reassemble the enclosure and plug in the USB cable and plug it into your Mac.
For the 2015 MBA (per your comments), you will need this enclosure. The 2010, 2012 and 2013-15 MBA's all use completely different interfaces. Make sure you select the one that is for your generation MBA.
You can then mount the drive and transfer your data.